Spot Rate Quotations

The spot rate is the interest that prevails today for a loan made today with a maturity oftyears. It will be denotedi0,t. The first index (0 here) refers to the time at which the loan starts; the second index (t) designates the number of years during which the loan will be running. We should warn the reader that it is not unusual to encounter other notations for interest rates bearing three indexes rather than our two. Sometimes interest rates are referred to asi0,0,t; the first index (0) refers to the time at which the decision to make the loan is made; the second index (0) refers to the start-up time of the loan; and the third index (t) may be either the date at which the loan is due or the number of years during which the loan will be running. In this text, since we will practically always deal with interest rate contracts that are decided today that come into existence either immediately or at some future date, for simplicity we have decided to suppress the first of the above three indexes.

The spot rate is, in effect, the yield to maturity of a zero-coupon bond with maturityt, becausei0,tverifies the equation

(1)

whereB0is the value today of a zero-coupon bond,tis its maturity, andBtis its par (or reimbursement) value. We could also say, equivalently, thati0,tis the horizon rate of return of a zero-coupon bond bought today at priceB0and having an expected—and certain—value ofBtat timet. From (1), we can extract the horizon rate of return, as we have done in earlier chapters, and obtain

(2)

So the concept of spot rate is entirely equivalent to the concept of horizon rate of return.Dividing both sides of (1) by (1 +i0,t)t, we can write thati0,tis such that

(3)

We deduce from (3) that the spot rate is the internal rate of return of the investment, derived from buying a bondB0and receivingBttyears later. Alternatively, we can see that the spot ratei0,tis the rate that makes the net present value of that project equal to zero:

We conclude that the spot rate is the internal rate of the investment project derived from buying the zero-coupon bond, or equivalently the yield to maturity of the zero-coupon bond. The following table summarizes these properties of the spot rate.

Equivalent Definitions

Spot ratei0,tfor a loan starting at time 0 and in effect fortyears

Horizon rate of return of an investment transforming an investmentB0intoBttyears later:

The term structure of interest rates is the structure of spot rates i0,tfor all maturities t. It is important not to confuse this with the yield curve. How can we know the spot rate curve i0,t? The easiest way to determine this information would be to calculate the horizon rates of return for a whole series of defaultless zero-coupon bonds; this would give us our result immediately. Suppose that B0,tdesignates the price today of a zero-coupon bond that will be reimbursed at Btin t years. We would simply have to calculate i0,t= (Bt/ B0,t)1/ t− 1 for the whole series of observed prices B0,t, and we would be done. Unfortunately, markets do not carry many zero-coupon bonds, especially in the range of medium to long maturities. Therefore, we have to infer, or calculate, the spot rates by relying on the values of coupon-bearing bonds

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Summary of the Trade-off Models

The fact that the same capital structure both minimizes the cost of capital and maximizes value should be no surprise. The value of any business is nothing more than the present value of its expected after-tax operating income stream. What discount rate is used to find the present value? It is the corporate cost of capital. Therefore, by minimizing its corporate cost of capital, a business is automatically creating the greatest value.

Unfortunately, it is extremely difficult for financial managers to actually quantify the costs and benefits of debt financing to their firms, so it is virtually impossible to pinpoint the capital structure that truly maximizes a business's value. Most experts believe that such a structure exists for every taxable business but that it changes substantially over time as the nature of the business and the capital markets changes. Most experts also believe that, as shown in the lower graph of Figure 10.3, the relationship between firm value and leverage is relatively flat; thus, relatively large deviations from the optimal structure can occur without materially affecting a business's value.

Now, consider the asymmetric information model. Because of asymmetric information, investors know less about a firm's prospects than do itsmanagers. Furthermore, managers try to maximize value for current stockholders, not new ones, so if the firm has excellent prospects, management will not want to issue new shares, but if things look bleak, then a new stock offering may be sold. Therefore, investors take a stock offering to be a signal of bad news, so stock prices tend to decline when new issues are announced. As a result, new equity financing can be very expensive, and this fact must be incorporated into the capital structure decision. Its effect is to motivate firms to maintain a reserve borrowing capacity, which permits future investment opportunities to be financed by debt when internal funds are insufficient.

By combining the two theories, we obtain this possible explanation for the capital structure decisions of taxable firms:

· Debt financing provides benefits because of the tax deductibility of interest. Hence, firms should have some debt in their capital structures.

· However, financial distress costs place limits on debt usage—beyond some point, these costs offset the tax advantage of debt.

· Finally, because of asymmetric information, businesses maintain a reserve borrowing capacity to take advantage of good investment opportunities and, at the same time, avoid having to issue stock at distressed prices.